On Cox–Kemperman moment inequalities for independent centered random variables

In 1983 Cox and Kemperman proved that Ef(ξ)+Ef(η)≤Ef(ξ+η) for functions ff with convex second derivative and independent centered random variables ξξ and ηη. We suggest another proof, show that the minimal moment restrictions are sufficient, and write out a less restrictive condition on ff for the inequality to hold.